Fit log returns to F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of
freedom).
xi is the estimated skewness parameter.
The long version of Velliv medium risk data runs from January 2007 to
April 2024 (incl).
For January 2007 to May 2012 no low risk and high risk funds existed.
For this period the medium risk data is copied into the other funds.
The short version runs from June 2012 to April 2024.
Velliv returns are including bonus and “DinKapital.
PFA returns are including”KundeKapital”.
The summary statistics are transformed back to the scale of gross returns by taking \(exp()\) of each summary statistic. (Note: Taking arithmetic mean of gross returns directly is no good. Must be geometric mean.)
| vmr | vhr | vmrl | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|---|
| Min. : | 0.901 | 0.877 | 0.901 | 0.924 | 0.886 | 0.912 | 0.882 | 0.893 | 0.899 |
| 1st Qu.: | 0.996 | 0.994 | 0.995 | 0.999 | 0.994 | 0.997 | 0.995 | 0.996 | 0.996 |
| Median : | 1.010 | 1.012 | 1.007 | 1.008 | 1.012 | 1.009 | 1.013 | 1.011 | 1.011 |
| Mean : | 1.006 | 1.007 | 1.005 | 1.005 | 1.008 | 1.006 | 1.008 | 1.007 | 1.006 |
| 3rd Qu.: | 1.021 | 1.027 | 1.020 | 1.015 | 1.025 | 1.018 | 1.025 | 1.022 | 1.021 |
| Max. : | 1.070 | 1.088 | 1.070 | 1.043 | 1.079 | 1.054 | 1.082 | 1.073 | 1.065 |
| Min. : | ranking | 1st Qu.: | ranking | Median : | ranking | Mean : | ranking | 3rd Qu.: | ranking | Max. : | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.924 | pmr | 0.999 | pmr | 1.013 | mhr | 1.008 | phr | 1.027 | vhr | 1.088 | vhr |
| 0.912 | mmr | 0.997 | mmr | 1.012 | phr | 1.008 | mhr | 1.025 | phr | 1.082 | mhr |
| 0.901 | vmr | 0.996 | vmr | 1.012 | vhr | 1.007 | vhr | 1.025 | mhr | 1.079 | phr |
| 0.901 | vmrl | 0.996 | vhr_pmr | 1.011 | vmr_phr | 1.007 | vmr_phr | 1.022 | vmr_phr | 1.073 | vmr_phr |
| 0.899 | vhr_pmr | 0.996 | vmr_phr | 1.011 | vhr_pmr | 1.006 | vhr_pmr | 1.021 | vmr | 1.070 | vmr |
| 0.893 | vmr_phr | 0.995 | mhr | 1.010 | vmr | 1.006 | vmr | 1.021 | vhr_pmr | 1.070 | vmrl |
| 0.886 | phr | 0.995 | vmrl | 1.009 | mmr | 1.006 | mmr | 1.020 | vmrl | 1.065 | vhr_pmr |
| 0.882 | mhr | 0.994 | phr | 1.008 | pmr | 1.005 | pmr | 1.018 | mmr | 1.054 | mmr |
| 0.877 | vhr | 0.994 | vhr | 1.007 | vmrl | 1.005 | vmrl | 1.015 | pmr | 1.043 | pmr |
Correlations
| vmr | vhr | pmr | phr | |
|---|---|---|---|---|
| vmr | 1.000 | 0.997 | 0.961 | 0.964 |
| vhr | 0.997 | 1.000 | 0.951 | 0.967 |
| pmr | 0.961 | 0.951 | 1.000 | 0.977 |
| phr | 0.964 | 0.967 | 0.977 | 1.000 |
Covariances
| vmr | vhr | pmr | phr | |
|---|---|---|---|---|
| vmr | 0.001 | 0.001 | 0 | 0.001 |
| vhr | 0.001 | 0.001 | 0 | 0.001 |
| pmr | 0.000 | 0.000 | 0 | 0.000 |
| phr | 0.001 | 0.001 | 0 | 0.001 |
Risk of loss at least as big as row name in percent for a single period (year).
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 32.333 | 33.000 | 30.167 | 31.667 | 31.167 | 32.167 | 31.833 | 31.667 |
| 5 | 2.500 | 4.167 | 0.667 | 3.167 | 1.500 | 3.500 | 2.667 | 2.500 |
| 10 | 0.167 | 0.500 | 0.000 | 0.167 | 0.000 | 0.333 | 0.167 | 0.167 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 90 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 99 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 28.833 | 29.500 | 22.667 | 29.333 | 25.667 | 29.333 | 29 | 27.333 |
| 5 | 1.833 | 3.167 | 0.833 | 2.500 | 1.167 | 2.667 | 2 | 1.833 |
| 10 | 0.000 | 0.333 | 0.000 | 0.000 | 0.000 | 0.167 | 0 | 0.000 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 |
| 90 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 |
| 99 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 40.333 | 40.667 | 37.833 | 38.500 | 39.167 | 39.500 | 39.167 | 39.667 |
| 5 | 0.500 | 2.333 | 0.000 | 1.333 | 0.000 | 1.667 | 0.833 | 0.500 |
| 10 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 90 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 99 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Skewed \(t\)-distribution (sstd):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.000 | vhr | 4.167 | vhr | 0.500 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 32.333 | vmr | 3.500 | mhr | 0.333 | mhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 32.167 | mhr | 3.167 | phr | 0.167 | vmr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 31.833 | vmr_phr | 2.667 | vmr_phr | 0.167 | phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 31.667 | phr | 2.500 | vmr | 0.167 | vmr_phr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 31.667 | vhr_pmr | 2.500 | vhr_pmr | 0.167 | vhr_pmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 31.167 | mmr | 1.500 | mmr | 0.000 | pmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 30.167 | pmr | 0.667 | pmr | 0.000 | mmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Standardized \(t\)-distribution (std):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 29.500 | vhr | 3.167 | vhr | 0.333 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 29.333 | phr | 2.667 | mhr | 0.167 | mhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 29.333 | mhr | 2.500 | phr | 0.000 | vmr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 29.000 | vmr_phr | 2.000 | vmr_phr | 0.000 | pmr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 28.833 | vmr | 1.833 | vmr | 0.000 | phr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 27.333 | vhr_pmr | 1.833 | vhr_pmr | 0.000 | mmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 25.667 | mmr | 1.167 | mmr | 0.000 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 22.667 | pmr | 0.833 | pmr | 0.000 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Normal distribution:
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 40.667 | vhr | 2.333 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 40.333 | vmr | 1.667 | mhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 39.667 | vhr_pmr | 1.333 | phr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 39.500 | mhr | 0.833 | vmr_phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 39.167 | mmr | 0.500 | vmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 39.167 | vmr_phr | 0.500 | vhr_pmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 38.500 | phr | 0.000 | pmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 37.833 | pmr | 0.000 | mmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Chance of gains of at least x percent for a single
period (year).
x values are row names.
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 67.667 | 67.000 | 69.833 | 68.333 | 68.833 | 67.833 | 68.167 | 68.333 |
| 5 | 1.167 | 3.833 | 0.167 | 3.667 | 0.500 | 3.333 | 2.167 | 1.333 |
| 10 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 71.167 | 70.500 | 77.333 | 70.667 | 74.333 | 70.667 | 71.000 | 72.667 |
| 5 | 7.833 | 13.500 | 3.667 | 11.500 | 5.500 | 12.500 | 9.667 | 8.167 |
| 10 | 0.667 | 1.833 | 0.167 | 1.167 | 0.333 | 1.333 | 0.833 | 0.833 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 59.667 | 59.333 | 62.167 | 61.500 | 60.833 | 60.5 | 60.833 | 60.333 |
| 5 | 3.333 | 8.333 | 0.167 | 7.167 | 1.333 | 7.5 | 5.167 | 3.500 |
| 10 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0 | 0.000 | 0.000 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0 | 0.000 | 0.000 |
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0 | 0.000 | 0.000 |
Skewed \(t\)-distribution (sstd):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 69.833 | pmr | 3.833 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 68.833 | mmr | 3.667 | phr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 68.333 | phr | 3.333 | mhr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 68.333 | vhr_pmr | 2.167 | vmr_phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 68.167 | vmr_phr | 1.333 | vhr_pmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 67.833 | mhr | 1.167 | vmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 67.667 | vmr | 0.500 | mmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 67.000 | vhr | 0.167 | pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Standardized \(t\)-distribution (std):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 77.333 | pmr | 13.500 | vhr | 1.833 | vhr | 0 | vmr | 0 | vmr | 0 | vmr |
| 74.333 | mmr | 12.500 | mhr | 1.333 | mhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 72.667 | vhr_pmr | 11.500 | phr | 1.167 | phr | 0 | pmr | 0 | pmr | 0 | pmr |
| 71.167 | vmr | 9.667 | vmr_phr | 0.833 | vmr_phr | 0 | phr | 0 | phr | 0 | phr |
| 71.000 | vmr_phr | 8.167 | vhr_pmr | 0.833 | vhr_pmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 70.667 | phr | 7.833 | vmr | 0.667 | vmr | 0 | mhr | 0 | mhr | 0 | mhr |
| 70.667 | mhr | 5.500 | mmr | 0.333 | mmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 70.500 | vhr | 3.667 | pmr | 0.167 | pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Normal distribution:
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 62.167 | pmr | 8.333 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 61.500 | phr | 7.500 | mhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 60.833 | mmr | 7.167 | phr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 60.833 | vmr_phr | 5.167 | vmr_phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 60.500 | mhr | 3.500 | vhr_pmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 60.333 | vhr_pmr | 3.333 | vmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 59.667 | vmr | 1.333 | mmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 59.333 | vhr | 0.167 | pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Risk of loss at least as big as row name in percent from first to last period.
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 17.71 | 18.08 | 10.00 | 11.89 | 7.65 | 7.23 | 7.11 | 8.27 |
| 5 | 9.25 | 11.04 | 3.51 | 6.44 | 2.25 | 2.58 | 2.60 | 2.78 |
| 10 | 4.54 | 6.57 | 1.19 | 3.42 | 0.62 | 0.90 | 0.75 | 0.85 |
| 25 | 0.47 | 0.88 | 0.09 | 0.35 | 0.02 | 0.02 | 0.04 | 0.03 |
| 50 | 0.02 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 4.21 | 4.55 | 2.33 | 3.44 | 0.64 | 0.58 | 0.60 | 0.62 |
| 5 | 2.12 | 2.64 | 1.13 | 1.67 | 0.30 | 0.18 | 0.18 | 0.23 |
| 10 | 1.08 | 1.32 | 0.57 | 0.73 | 0.11 | 0.11 | 0.08 | 0.06 |
| 25 | 0.10 | 0.21 | 0.15 | 0.06 | 0.02 | 0.01 | 0.00 | 0.01 |
| 50 | 0.00 | 0.00 | 0.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 90 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 14.05 | 15.07 | 9.03 | 9.83 | 4.83 | 4.39 | 4.46 | 5.75 |
| 5 | 6.01 | 8.08 | 2.22 | 4.34 | 0.64 | 1.03 | 1.01 | 1.15 |
| 10 | 1.97 | 3.53 | 0.29 | 1.66 | 0.02 | 0.23 | 0.14 | 0.12 |
| 25 | 0.00 | 0.07 | 0.00 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 |
| 50 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Skewed \(t\)-distribution (sstd):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.08 | vhr | 11.04 | vhr | 6.57 | vhr | 0.88 | vhr | 0.03 | vhr | 0 | vmr | 0 | vmr |
| 17.71 | vmr | 9.25 | vmr | 4.54 | vmr | 0.47 | vmr | 0.02 | vmr | 0 | vhr | 0 | vhr |
| 11.89 | phr | 6.44 | phr | 3.42 | phr | 0.35 | phr | 0.01 | pmr | 0 | pmr | 0 | pmr |
| 10.00 | pmr | 3.51 | pmr | 1.19 | pmr | 0.09 | pmr | 0.00 | phr | 0 | phr | 0 | phr |
| 8.27 | vhr_pmr | 2.78 | vhr_pmr | 0.90 | mhr | 0.04 | vmr_phr | 0.00 | mmr | 0 | mmr | 0 | mmr |
| 7.65 | mmr | 2.60 | vmr_phr | 0.85 | vhr_pmr | 0.03 | vhr_pmr | 0.00 | mhr | 0 | mhr | 0 | mhr |
| 7.23 | mhr | 2.58 | mhr | 0.75 | vmr_phr | 0.02 | mmr | 0.00 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 7.11 | vmr_phr | 2.25 | mmr | 0.62 | mmr | 0.02 | mhr | 0.00 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Standardized \(t\)-distribution (std):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.55 | vhr | 2.64 | vhr | 1.32 | vhr | 0.21 | vhr | 0.04 | pmr | 0.01 | pmr | 0 | vmr |
| 4.21 | vmr | 2.12 | vmr | 1.08 | vmr | 0.15 | pmr | 0.00 | vmr | 0.00 | vmr | 0 | vhr |
| 3.44 | phr | 1.67 | phr | 0.73 | phr | 0.10 | vmr | 0.00 | vhr | 0.00 | vhr | 0 | pmr |
| 2.33 | pmr | 1.13 | pmr | 0.57 | pmr | 0.06 | phr | 0.00 | phr | 0.00 | phr | 0 | phr |
| 0.64 | mmr | 0.30 | mmr | 0.11 | mmr | 0.02 | mmr | 0.00 | mmr | 0.00 | mmr | 0 | mmr |
| 0.62 | vhr_pmr | 0.23 | vhr_pmr | 0.11 | mhr | 0.01 | mhr | 0.00 | mhr | 0.00 | mhr | 0 | mhr |
| 0.60 | vmr_phr | 0.18 | mhr | 0.08 | vmr_phr | 0.01 | vhr_pmr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0 | vmr_phr |
| 0.58 | mhr | 0.18 | vmr_phr | 0.06 | vhr_pmr | 0.00 | vmr_phr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0 | vhr_pmr |
Normal distribution:
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15.07 | vhr | 8.08 | vhr | 3.53 | vhr | 0.07 | vhr | 0 | vmr | 0 | vmr | 0 | vmr |
| 14.05 | vmr | 6.01 | vmr | 1.97 | vmr | 0.02 | phr | 0 | vhr | 0 | vhr | 0 | vhr |
| 9.83 | phr | 4.34 | phr | 1.66 | phr | 0.00 | vmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 9.03 | pmr | 2.22 | pmr | 0.29 | pmr | 0.00 | pmr | 0 | phr | 0 | phr | 0 | phr |
| 5.75 | vhr_pmr | 1.15 | vhr_pmr | 0.23 | mhr | 0.00 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 4.83 | mmr | 1.03 | mhr | 0.14 | vmr_phr | 0.00 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 4.46 | vmr_phr | 1.01 | vmr_phr | 0.12 | vhr_pmr | 0.00 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 4.39 | mhr | 0.64 | mmr | 0.02 | mmr | 0.00 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 82.29 | 81.92 | 90.00 | 88.11 | 92.35 | 92.77 | 92.89 | 91.73 |
| 5 | 70.65 | 72.69 | 75.97 | 79.76 | 78.86 | 84.34 | 82.99 | 80.04 |
| 10 | 55.47 | 61.27 | 55.27 | 68.75 | 55.72 | 70.44 | 66.97 | 61.66 |
| 25 | 13.20 | 24.67 | 4.82 | 29.65 | 3.28 | 20.10 | 13.41 | 9.13 |
| 50 | 0.14 | 1.52 | 0.02 | 1.64 | 0.00 | 0.21 | 0.02 | 0.05 |
| 100 | 0.00 | 0.04 | 0.01 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 |
| 200 | 0.00 | 0.01 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 |
| 300 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 400 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 500 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 1000 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 95.79 | 95.45 | 97.67 | 96.56 | 99.36 | 99.42 | 99.40 | 99.38 |
| 5 | 92.09 | 92.35 | 94.81 | 93.66 | 98.00 | 98.15 | 98.21 | 98.12 |
| 10 | 86.07 | 87.80 | 89.06 | 88.99 | 94.76 | 95.63 | 94.87 | 95.25 |
| 25 | 54.92 | 65.59 | 49.09 | 64.12 | 54.65 | 72.05 | 64.24 | 65.03 |
| 50 | 10.26 | 24.55 | 4.50 | 18.98 | 2.96 | 14.88 | 7.98 | 8.18 |
| 100 | 0.28 | 1.31 | 0.17 | 0.50 | 0.05 | 0.12 | 0.04 | 0.11 |
| 200 | 0.03 | 0.04 | 0.02 | 0.00 | 0.01 | 0.00 | 0.00 | 0.01 |
| 300 | 0.02 | 0.02 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 400 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 500 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 85.95 | 84.93 | 90.97 | 90.17 | 95.17 | 95.61 | 95.54 | 94.25 |
| 5 | 72.53 | 74.85 | 75.33 | 81.58 | 82.78 | 87.77 | 85.46 | 82.92 |
| 10 | 56.07 | 62.19 | 52.17 | 69.19 | 58.26 | 74.22 | 69.32 | 62.77 |
| 25 | 13.98 | 25.16 | 4.79 | 29.60 | 4.08 | 21.63 | 13.74 | 9.72 |
| 50 | 0.40 | 2.19 | 0.00 | 2.08 | 0.00 | 0.21 | 0.05 | 0.01 |
| 100 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 200 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 300 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 400 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Skewed \(t\)-distribution (sstd):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 92.89 | vmr_phr | 84.34 | mhr | 70.44 | mhr | 29.65 | phr | 1.64 | phr | 0.04 | vhr |
| 92.77 | mhr | 82.99 | vmr_phr | 68.75 | phr | 24.67 | vhr | 1.52 | vhr | 0.01 | pmr |
| 92.35 | mmr | 80.04 | vhr_pmr | 66.97 | vmr_phr | 20.10 | mhr | 0.21 | mhr | 0.01 | phr |
| 91.73 | vhr_pmr | 79.76 | phr | 61.66 | vhr_pmr | 13.41 | vmr_phr | 0.14 | vmr | 0.00 | vmr |
| 90.00 | pmr | 78.86 | mmr | 61.27 | vhr | 13.20 | vmr | 0.05 | vhr_pmr | 0.00 | mmr |
| 88.11 | phr | 75.97 | pmr | 55.72 | mmr | 9.13 | vhr_pmr | 0.02 | pmr | 0.00 | mhr |
| 82.29 | vmr | 72.69 | vhr | 55.47 | vmr | 4.82 | pmr | 0.02 | vmr_phr | 0.00 | vmr_phr |
| 81.92 | vhr | 70.65 | vmr | 55.27 | pmr | 3.28 | mmr | 0.00 | mmr | 0.00 | vhr_pmr |
| 200 | ranking | 300 | ranking | 400 | ranking | 500 | ranking | 1000 | ranking |
|---|---|---|---|---|---|---|---|---|---|
| 0.01 | vhr | 0.01 | vhr | 0.01 | vhr | 0.01 | vhr | 0.01 | vhr |
| 0.01 | phr | 0.00 | vmr | 0.00 | vmr | 0.00 | vmr | 0.00 | vmr |
| 0.00 | vmr | 0.00 | pmr | 0.00 | pmr | 0.00 | pmr | 0.00 | pmr |
| 0.00 | pmr | 0.00 | phr | 0.00 | phr | 0.00 | phr | 0.00 | phr |
| 0.00 | mmr | 0.00 | mmr | 0.00 | mmr | 0.00 | mmr | 0.00 | mmr |
| 0.00 | mhr | 0.00 | mhr | 0.00 | mhr | 0.00 | mhr | 0.00 | mhr |
| 0.00 | vmr_phr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0.00 | vmr_phr |
| 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0.00 | vhr_pmr |
Standardized \(t\)-distribution (std):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.42 | mhr | 98.21 | vmr_phr | 95.63 | mhr | 72.05 | mhr | 24.55 | vhr | 1.31 | vhr |
| 99.40 | vmr_phr | 98.15 | mhr | 95.25 | vhr_pmr | 65.59 | vhr | 18.98 | phr | 0.50 | phr |
| 99.38 | vhr_pmr | 98.12 | vhr_pmr | 94.87 | vmr_phr | 65.03 | vhr_pmr | 14.88 | mhr | 0.28 | vmr |
| 99.36 | mmr | 98.00 | mmr | 94.76 | mmr | 64.24 | vmr_phr | 10.26 | vmr | 0.17 | pmr |
| 97.67 | pmr | 94.81 | pmr | 89.06 | pmr | 64.12 | phr | 8.18 | vhr_pmr | 0.12 | mhr |
| 96.56 | phr | 93.66 | phr | 88.99 | phr | 54.92 | vmr | 7.98 | vmr_phr | 0.11 | vhr_pmr |
| 95.79 | vmr | 92.35 | vhr | 87.80 | vhr | 54.65 | mmr | 4.50 | pmr | 0.05 | mmr |
| 95.45 | vhr | 92.09 | vmr | 86.07 | vmr | 49.09 | pmr | 2.96 | mmr | 0.04 | vmr_phr |
| 200 | ranking | 300 | ranking | 400 | ranking | 500 | ranking | 1000 | ranking |
|---|---|---|---|---|---|---|---|---|---|
| 0.04 | vhr | 0.02 | vmr | 0.01 | pmr | 0.01 | pmr | 0 | vmr |
| 0.03 | vmr | 0.02 | vhr | 0.00 | vmr | 0.00 | vmr | 0 | vhr |
| 0.02 | pmr | 0.02 | pmr | 0.00 | vhr | 0.00 | vhr | 0 | pmr |
| 0.01 | mmr | 0.00 | phr | 0.00 | phr | 0.00 | phr | 0 | phr |
| 0.01 | vhr_pmr | 0.00 | mmr | 0.00 | mmr | 0.00 | mmr | 0 | mmr |
| 0.00 | phr | 0.00 | mhr | 0.00 | mhr | 0.00 | mhr | 0 | mhr |
| 0.00 | mhr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0 | vmr_phr |
| 0.00 | vmr_phr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0 | vhr_pmr |
Normal distribution:
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 95.61 | mhr | 87.77 | mhr | 74.22 | mhr | 29.60 | phr | 2.19 | vhr | 0 | vmr |
| 95.54 | vmr_phr | 85.46 | vmr_phr | 69.32 | vmr_phr | 25.16 | vhr | 2.08 | phr | 0 | vhr |
| 95.17 | mmr | 82.92 | vhr_pmr | 69.19 | phr | 21.63 | mhr | 0.40 | vmr | 0 | pmr |
| 94.25 | vhr_pmr | 82.78 | mmr | 62.77 | vhr_pmr | 13.98 | vmr | 0.21 | mhr | 0 | phr |
| 90.97 | pmr | 81.58 | phr | 62.19 | vhr | 13.74 | vmr_phr | 0.05 | vmr_phr | 0 | mmr |
| 90.17 | phr | 75.33 | pmr | 58.26 | mmr | 9.72 | vhr_pmr | 0.01 | vhr_pmr | 0 | mhr |
| 85.95 | vmr | 74.85 | vhr | 56.07 | vmr | 4.79 | pmr | 0.00 | pmr | 0 | vmr_phr |
| 84.93 | vhr | 72.53 | vmr | 52.17 | pmr | 4.08 | mmr | 0.00 | mmr | 0 | vhr_pmr |
| 200 | ranking | 300 | ranking | 400 | ranking | 500 | ranking | 1000 | ranking |
|---|---|---|---|---|---|---|---|---|---|
| 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 0 | phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Summary for fit of log returns to an F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape
parameter.
xi is the estimated skewness parameter.
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| m | 0.005 | 0.007 | 0.005 | 0.008 | 0.005 | 0.007 | 0.007 | 0.006 |
| s | 0.027 | 0.034 | 0.019 | 0.030 | 0.023 | 0.031 | 0.028 | 0.027 |
| nu | 3.384 | 3.488 | 3.474 | 3.959 | 3.344 | 3.702 | 3.726 | 3.369 |
| xi | 0.699 | 0.708 | 0.770 | 0.737 | 0.716 | 0.714 | 0.715 | 0.709 |
| R^2 | 0.993 | 0.992 | 0.994 | 0.996 | 0.993 | 0.993 | 0.994 | 0.993 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| m | 0.013 | 0.015 | 0.012 | 0.014 | 0.012 | 0.015 | 0.014 | 0.013 |
| s | 0.032 | 0.040 | 0.027 | 0.035 | 0.029 | 0.037 | 0.033 | 0.033 |
| nu | 3.446 | 3.510 | 2.629 | 4.002 | 3.035 | 3.780 | 3.760 | 3.260 |
| R^2 | 0.978 | 0.978 | 0.962 | 0.981 | 0.971 | 0.977 | 0.978 | 0.974 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| m | 0.006 | 0.007 | 0.005 | 0.008 | 0.006 | 0.008 | 0.007 | 0.006 |
| s | 0.024 | 0.031 | 0.017 | 0.028 | 0.021 | 0.029 | 0.026 | 0.024 |
| R^2 | 0.968 | 0.969 | 0.962 | 0.973 | 0.965 | 0.969 | 0.969 | 0.966 |
AIC
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| sstd | -671.412 | -603.343 | -768.368 | -622.474 | -718.914 | -617.207 | -647.901 | -672.555 |
| std | -628.069 | -561.055 | -728.782 | -585.156 | -674.537 | -574.620 | -605.443 | -628.239 |
| normal | -646.514 | -579.369 | -743.179 | -603.088 | -692.459 | -593.830 | -624.670 | -646.870 |
BIC
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| sstd | -659.588 | -591.520 | -756.545 | -610.651 | -707.091 | -605.384 | -636.077 | -660.732 |
| std | -616.246 | -549.232 | -716.959 | -573.333 | -662.714 | -562.796 | -593.620 | -616.416 |
| normal | -634.691 | -567.546 | -731.356 | -591.265 | -680.636 | -582.006 | -612.847 | -635.046 |
Let \(\{X_{g,i}\}\) be Gaussian distributed with mean \(\mu\) and scale \(\sigma\).
Let \(\{X_{\nu,i}\}\) be \(t\)-distributed, scaled such that \(\mathbb{M}^{\nu}(1) = \mathbb{M}^{g}(1) = \sqrt{\frac{2}{\pi}} \sigma\).
Given \(n_g\), we want to determine and \(n_{\nu}^{*}\) such that
\[\text{Var}\left[\sum_i^{n_g} X_{g,i}\right] = \text{Var}\left[\sum_i^{n_{\nu}^{*}} X_{\nu,i}\right]\]
For iid. r.v \(\{X_i\}\):
\[S_n = X_1 + X_2 + \dots + X_n\] \[\mathbb{M}(n) = \mathbb{E}(\lvert S_n - \mathbb{E}(S_n)\rvert)\] Taleb’s convergence metric (\(\kappa\)):
The “rate” of convergence for \(n\) summands vs \(n_0\), i.e. the improved convergence achieved by \(n - n_0\) additional terms, is given by \(\kappa(n_0, n)\):
\[\kappa(n_0, n) = 2 - \dfrac{\log(n) - \log(n_0)}{\log\left(\frac{\mathbb{M}(n)}{\mathbb{M}(n_0)}\right)}\]
\(\kappa\)
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr |
|---|---|---|---|---|---|---|---|
| 0.16 | 0.16 | 0.16 | 0.13 | 0.15 | 0.14 | 0.13 | 0.17 |
\(n_{min}\)
What is the minimum value of \(n_{\nu}\), the number of observations from a given skewed \(t\)-distribution, we need to achieve the same degree of convergence as with \(n_g=30\) observations from a Gaussian distribution with the same mean and standard deviation?
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr |
|---|---|---|---|---|---|---|---|
| 58 | 55 | 55 | 48 | 58 | 51 | 50 | 57 |
Skewed \(t\)-distribution (sstd):
| m | ranking | s | ranking | R^2 | ranking |
|---|---|---|---|---|---|
| 0.008 | phr | 0.019 | pmr | 0.996 | phr |
| 0.007 | mhr | 0.023 | mmr | 0.994 | vmr_phr |
| 0.007 | vmr_phr | 0.027 | vhr_pmr | 0.994 | pmr |
| 0.007 | vhr | 0.027 | vmr | 0.993 | mmr |
| 0.006 | vhr_pmr | 0.028 | vmr_phr | 0.993 | mhr |
| 0.005 | vmr | 0.030 | phr | 0.993 | vmr |
| 0.005 | pmr | 0.031 | mhr | 0.993 | vhr_pmr |
| 0.005 | mmr | 0.034 | vhr | 0.992 | vhr |
Standardized \(t\)-distribution (std):
| m | ranking | s | ranking | R^2 | ranking |
|---|---|---|---|---|---|
| 0.015 | vhr | 0.027 | pmr | 0.981 | phr |
| 0.015 | mhr | 0.029 | mmr | 0.978 | vmr |
| 0.014 | phr | 0.032 | vmr | 0.978 | vhr |
| 0.014 | vmr_phr | 0.033 | vhr_pmr | 0.978 | vmr_phr |
| 0.013 | vhr_pmr | 0.033 | vmr_phr | 0.977 | mhr |
| 0.013 | vmr | 0.035 | phr | 0.974 | vhr_pmr |
| 0.012 | mmr | 0.037 | mhr | 0.971 | mmr |
| 0.012 | pmr | 0.040 | vhr | 0.962 | pmr |
Normal distribution:
| m | ranking | s | ranking | R^2 | ranking |
|---|---|---|---|---|---|
| 0.008 | phr | 0.017 | pmr | 0.973 | phr |
| 0.008 | mhr | 0.021 | mmr | 0.969 | vmr_phr |
| 0.007 | vhr | 0.024 | vhr_pmr | 0.969 | vhr |
| 0.007 | vmr_phr | 0.024 | vmr | 0.969 | mhr |
| 0.006 | vhr_pmr | 0.026 | vmr_phr | 0.968 | vmr |
| 0.006 | vmr | 0.028 | phr | 0.966 | vhr_pmr |
| 0.006 | mmr | 0.029 | mhr | 0.965 | mmr |
| 0.005 | pmr | 0.031 | vhr | 0.962 | pmr |
Monte Carlo simulations of portfolio index values (currency
values).
Statistics are given for the final state of all paths.
Probability of down-and-out is calculated as the share of paths that
reach 0 at some point. All subsequent values for a path are set to 0, if
the path reaches at any point.
0 is defined as any value below a threshold.
dai_pct (for down-and-in) is the probability of losing
money. This is calculated as the share of paths finishing below index
100.
## Number of paths: 10000
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| mc_m | 390.56 | 546.54 | 368.99 | 744.90 | 379.50 | 649.19 | 569.75 | 458.13 |
| mc_s | 162.67 | 310.60 | 106.60 | 358.90 | 98.41 | 230.77 | 195.66 | 154.95 |
| mc_min | 9.95 | 47.50 | 41.88 | 67.17 | 120.88 | 157.64 | 134.70 | 146.99 |
| mc_max | 1962.47 | 9418.03 | 964.46 | 3790.71 | 969.20 | 2350.77 | 2262.60 | 1819.14 |
| dao_pct | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| dai_pct | 0.32 | 0.32 | 0.03 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| mc_m | 2281.79 | 4769.87 | 1795.80 | 3664.55 | 2345.79 | 4253.11 | 2979.65 | 3285.27 |
| mc_s | 1214.65 | 3240.61 | 1239.38 | 2099.78 | 31176.36 | 3565.88 | 1263.15 | 1921.35 |
| mc_min | 221.36 | 374.70 | 41.43 | 193.68 | 557.15 | 783.35 | 689.69 | 499.82 |
| mc_max | 16109.79 | 45965.08 | 94908.01 | 28483.53 | 3118787.05 | 295755.16 | 30086.97 | 59511.22 |
| dao_pct | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| dai_pct | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| mc_m | 441.92 | 627.78 | 365.42 | 789.64 | 403.75 | 711.18 | 614.07 | 498.41 |
| mc_s | 169.65 | 319.17 | 98.48 | 362.10 | 98.81 | 244.56 | 197.97 | 164.59 |
| mc_min | 90.51 | 80.75 | 123.03 | 120.90 | 165.90 | 188.76 | 136.86 | 160.53 |
| mc_max | 1791.46 | 3370.41 | 1109.65 | 3404.35 | 1102.37 | 2538.96 | 1961.78 | 1606.02 |
| dao_pct | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| dai_pct | 0.01 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Skewed \(t\)-distribution (sstd):
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_pct | ranking | dai_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 744.90 | phr | 98.41 | mmr | 157.64 | mhr | 9418.03 | vhr | 0 | vmr | 0.00 | mmr |
| 649.19 | mhr | 106.60 | pmr | 146.99 | vhr_pmr | 3790.71 | phr | 0 | vhr | 0.00 | mhr |
| 569.75 | vmr_phr | 154.95 | vhr_pmr | 134.70 | vmr_phr | 2350.77 | mhr | 0 | pmr | 0.00 | vmr_phr |
| 546.54 | vhr | 162.67 | vmr | 120.88 | mmr | 2262.60 | vmr_phr | 0 | phr | 0.00 | vhr_pmr |
| 458.13 | vhr_pmr | 195.66 | vmr_phr | 67.17 | phr | 1962.47 | vmr | 0 | mmr | 0.03 | pmr |
| 390.56 | vmr | 230.77 | mhr | 47.50 | vhr | 1819.14 | vhr_pmr | 0 | mhr | 0.03 | phr |
| 379.50 | mmr | 310.60 | vhr | 41.88 | pmr | 969.20 | mmr | 0 | vmr_phr | 0.32 | vmr |
| 368.99 | pmr | 358.90 | phr | 9.95 | vmr | 964.46 | pmr | 0 | vhr_pmr | 0.32 | vhr |
Standardized \(t\)-distribution (std):
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_pct | ranking | dai_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 4769.87 | vhr | 1214.65 | vmr | 783.35 | mhr | 3118787.05 | mmr | 0 | vmr | 0.00 | vmr |
| 4253.11 | mhr | 1239.38 | pmr | 689.69 | vmr_phr | 295755.16 | mhr | 0 | vhr | 0.00 | vhr |
| 3664.55 | phr | 1263.15 | vmr_phr | 557.15 | mmr | 94908.01 | pmr | 0 | pmr | 0.00 | phr |
| 3285.27 | vhr_pmr | 1921.35 | vhr_pmr | 499.82 | vhr_pmr | 59511.22 | vhr_pmr | 0 | phr | 0.00 | mmr |
| 2979.65 | vmr_phr | 2099.78 | phr | 374.70 | vhr | 45965.08 | vhr | 0 | mmr | 0.00 | mhr |
| 2345.79 | mmr | 3240.61 | vhr | 221.36 | vmr | 30086.97 | vmr_phr | 0 | mhr | 0.00 | vmr_phr |
| 2281.79 | vmr | 3565.88 | mhr | 193.68 | phr | 28483.53 | phr | 0 | vmr_phr | 0.00 | vhr_pmr |
| 1795.80 | pmr | 31176.36 | mmr | 41.43 | pmr | 16109.79 | vmr | 0 | vhr_pmr | 0.01 | pmr |
Normal distribution:
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_pct | ranking | dai_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 789.64 | phr | 98.48 | pmr | 188.76 | mhr | 3404.35 | phr | 0 | vmr | 0.00 | pmr |
| 711.18 | mhr | 98.81 | mmr | 165.90 | mmr | 3370.41 | vhr | 0 | vhr | 0.00 | phr |
| 627.78 | vhr | 164.59 | vhr_pmr | 160.53 | vhr_pmr | 2538.96 | mhr | 0 | pmr | 0.00 | mmr |
| 614.07 | vmr_phr | 169.65 | vmr | 136.86 | vmr_phr | 1961.78 | vmr_phr | 0 | phr | 0.00 | mhr |
| 498.41 | vhr_pmr | 197.97 | vmr_phr | 123.03 | pmr | 1791.46 | vmr | 0 | mmr | 0.00 | vmr_phr |
| 441.92 | vmr | 244.56 | mhr | 120.90 | phr | 1606.02 | vhr_pmr | 0 | mhr | 0.00 | vhr_pmr |
| 403.75 | mmr | 319.17 | vhr | 90.51 | vmr | 1109.65 | pmr | 0 | vmr_phr | 0.01 | vmr |
| 365.42 | pmr | 362.10 | phr | 80.75 | vhr | 1102.37 | mmr | 0 | vhr_pmr | 0.01 | vhr |
Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| P_norm(X_min) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| P_norm(X_max) | 0.546 | 0.580 | 1.599 | 0.796 | 1.161 | 0.746 | 0.793 | 0.903 |
| P_t(X_min) | 0.556 | 0.523 | 0.342 | 0.387 | 0.476 | 0.443 | 0.433 | 0.499 |
| P_t(X_max) | 0.448 | 0.469 | 1.135 | 0.614 | 0.739 | 0.518 | 0.543 | 0.613 |
Average number of years between min or max events (respectively):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| norm: avg yrs btw min | 1.157446e+77 | 3.872665e+82 | 3.449915e+57 | 4.076881e+55 | 2.913573e+67 | 1.048138e+67 | 2.446247e+63 | 4.894371e+71 |
| norm: avg yrs btw max | 1.830960e+02 | 1.724260e+02 | 6.252800e+01 | 1.256870e+02 | 8.614100e+01 | 1.341240e+02 | 1.261390e+02 | 1.107200e+02 |
| t: avg yrs btw min | 1.798360e+02 | 1.912300e+02 | 2.924120e+02 | 2.584920e+02 | 2.099400e+02 | 2.257430e+02 | 2.309140e+02 | 2.002190e+02 |
| t: avg yrs btw max | 2.233340e+02 | 2.130540e+02 | 8.811500e+01 | 1.628680e+02 | 1.352750e+02 | 1.928930e+02 | 1.843230e+02 | 1.630560e+02 |
p-values for Lilliefors test.
Testing \(H_0\), that log-returns are
Gaussian.
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| p value | 0 | 0 | 0 | 0 | 0 | 0.001 | 0.001 | 0 |
For different given probabilities that returns are Gaussian, what is the probability that the distribution is Gaussian rather than skewed t-distributed, given the smallest/largest observed log-returns?
Conditional probabilities for smallest observed log-returns:
Use \(1 - \text{p-value}\) from
Lilliefors test as prior probability that the distribution is
Gaussian.
\(x_{\text{obs}} = \min(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] =
P_{\text{Gauss}}[X \leq x_{\text{min}}]\):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| Lillie p-val | 0.000 | 0.00 | 0.000 | 0.00 | 0.000 | 0.001 | 0.001 | 0.000 |
| Prior prob | 1.000 | 1.00 | 1.000 | 1.00 | 1.000 | 0.999 | 0.999 | 1.000 |
| P[Gauss | Event] | 0.838 | 0.82 | 0.768 | 0.69 | 0.815 | 0.486 | 0.418 | 0.965 |
Use \(1 - \text{p-value}\) from
Lilliefors test as prior probability that the distribution is
Gaussian.
\(x_{\text{obs}} = \max(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] =
P_{\text{Gauss}}[X \geq x_{\text{max}}]\):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| Lillie p-val | 0 | 0 | 0 | 0 | 0 | 0.001 | 0.001 | 0 |
| Prior prob | 1 | 1 | 1 | 1 | 1 | 0.999 | 0.999 | 1 |
| P[Gauss | Event] | 1 | 1 | 1 | 1 | 1 | 1.000 | 1.000 | 1 |
Skewed \(t\)-distribution (sstd):
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Sorted portfolio index values for last period of all runs
Max vs sum plots for the first four moments:
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.4145605 0.3807834
Objective function plots
Skewed \(t\)-distribution (sstd):
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Sorted portfolio index values for last period of all runs
Max vs sum plots for the first four moments:
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.7391222 0.4858909
Objective function plots
Skewed \(t\)-distribution (sstd):
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Sorted portfolio index values for last period of all runs
Max vs sum plots for the first four moments:
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.3304634 0.2764028
Objective function plots
Skewed \(t\)-distribution (sstd):
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
phr has the sstd fit with the highest sstd fit with thevalue of nu. Compare with other distributions:
Sorted portfolio index values for last period of all runs
Max vs sum plots for the first four moments:
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 2.0162301 0.4463226
Objective function plots
Skewed \(t\)-distribution (sstd):
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
mmr has the sstd fit with the lowest value of nu. Compare with other distributions:
Sorted portfolio index values for last period of all runs
Max vs sum plots for the first four moments:
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.3516393 0.2503782
Objective function plots
Skewed \(t\)-distribution (sstd):
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Sorted portfolio index values for last period of all runs
Max vs sum plots for the first four moments:
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.8775189 0.3400818
Objective function plots
Skewed \(t\)-distribution (sstd):
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Sorted portfolio index values for last period of all runs
Max vs sum plots for the first four moments:
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.7507161 0.3263777
Objective function plots
Skewed \(t\)-distribution (sstd):
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Sorted portfolio index values for last period of all runs
Max vs sum plots for the first four moments:
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.540135 0.320090
Objective function plots
Skewed \(t\)-distribution (sstd):
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Sorted portfolio index values for last period of all runs
Max vs sum plots for the first four moments:
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.4145605 0.3807834
Objective function plots